Seeking Symmetry: Lie Group & Algebra Models

[fresh_42]

TL;DR Summary

Is there a physical system which uses these groups?

I wonder whether there is a physical theory / model / example whatever, that uses one of the (Lie) groups $\begin{bmatrix}1&a_2&\ldots&a_n\\0&1&\ldots&0\\ \vdots & \vdots &&\vdots \\0&0&\ldots&1 \end{bmatrix}$ or the Lie algebra $\begin{bmatrix}a_1&a_2&\ldots&a_n\\0&0&\ldots&0\\ \vdots & \vdots &&\vdots \\0&0&\ldots&0 \end{bmatrix}$?

 

[martinbn]

What exactly are the group and the algebra?

 

[Vanadium 50]

Under what operation is this a group? It can't be matrix multiplication because there is no identity.

 

[fresh_42]

Under what operation is this a group? It can't be matrix multiplication because there is no identity.

Silly me. Corrected.

What exactly are the group and the algebra?

Matrix multiplication and commutator, resp.

I would be happy if someone knew a representation of the Lie algebra in terms of differential operators.

 

[martinbn]

Then the group is just the additive group $\mathbb F^{n-1}$. The algebra should have zero in the top left, and is the trivial algebra, trivial commutator on $\mathbb F^{n-1}$.

 

[fresh_42]

No, the algebra needs the non zero value at top left. I want it to be non nilpotent with a trivial center. It's basically the simplest non trivial Lie algebra with the multiplication table $[X_1,X_i]=X_i$ and $[X_i.X_j]=0$. I admit that it's a bit of numerology I'm doing here. I'm chasing for an imagination of that Lie algebra. What is it good for? Does it occur anywhere? Modulo the fact that I didn't make any mistakes in my calculation, I found a cochain complex of that Lie algebra whose first cohomology group( space?) is $\mathfrak{sl}_{(n-1)}$. This is funny by itself, but I wonder if it is just that: funny. A realization or representation via diff operators would at least help me better understand this property: Simplest non Abelian Lie algebra.

 

[martinbn]

I assumed that the Lie algebra was supposed to be the Lie algebra of the given group. I ques that is not what you meant. In any case the group is just the additive group and is not that interesting. Of course translations appear here and there in physics.

 

[fresh_42]

It's becoming even more funnier: If we write $\mathfrak{g}_n :=\langle X_1,\ldots\, , \,X_n\,|\,[X-1,X_k]=X_k\, , \,[X_i,X_j]=0\rangle$ then I have a certain Chevalley-Eilenberg complex such that $H^0(\mathfrak{g}_n)=H^2(\mathfrak{g}_n)=\{0\}$ and $H^1(\mathfrak{g}_n)\cong\mathfrak{sl}(n-1)\cong \mathfrak{su}(n-1)$. That's why I asked whether there is any meaning for $\mathfrak{g}_n$. So far it's just an easy, but nasty calculation. I need a better understanding on the simple part: $\mathfrak{g}_n$.